Self-strutted geodesic plydome



Sept. 22, 1959 R. B. FULLER 2,905,113

SELF-STRUTTED GEODESIC PLYDOME Filed April 22. 1957 3 Sheets-Sheet 1INVENTOR. RICHARD BUCKMINSTER FULLER Armen/m. v

Sept. 22, 1959 R. B. FULLER SELF-STRUTTED GEODESIC PLYDOME 3Sheets-Sheet 2 Filed April 22. 1957 INVEN RICHARD BUCKMINST ER F ERSept. 22, 1959 R. B. FULLER 2,905,113 Y SELF-STRUTTED GEODESI PLYDOMEFiled April 22, 1957 I 5 Sheets-Sheet 5 i AMA@ En U5 n B IN VEN T0RICHARD BUCKMINSTER FUL ATTORNF sphere.

United States Patent Patented Sept. 22, 1959 iitice SELF-STRUTTEDGEODESIC PLYDOME Richard Buckminster Fuller, New York, N.Y.

Application April 22, 1957, Serial No. 654,166

15 Claims. (Cl. 10S-1) The invention relates to geodesic and synergeticconstruction of ydome-shaped enclosures.

Summary principal characteristics which distinguish it Ifrom the` olderarchitectural forms; so these characteristics will here be reviewed onlybriey. For a comprehensive review, reference is made to Patent No.2,682,235, aforesaid.

In geodesic construction, the building framework is` one of generallyspherical form in which the longitudinal centerlines of the mainstructural elements lie substantially in great circle planes whoseintersections with a common sphere form grids comprising substantiallyequilateral spherical triangles. [Great circle planes are defined asplanes whose yintersections with a sphere are great circles. Such planespass through the center of the The earths equator and the meridians ofthe globe are representative of great circles in the ordinary accepted`meaning of this term.] The grids can, for example, be formed on thefaces of a spherical icosahedron. Each of the twenty equal sphericalequilateral triangles which `form the faces of the icosa is modularlydivided along its edges. Lines connecting these modularly divided edgesin a three-way great circle grid provide the out-line for the plan ofconstruction. Each of the smaller triangles formed by the three-way gridis approximately equilateral, i.e. its sides are approximately equal.The extent off variation in length is determined tnigonometrically or bygraphic solution of the grids as drawn upon the modularly divided edgesof an icosahedron outlined upon the surface of a scale model sphere. Itwill be found that at each vertex of the icosa ve of the grid trianglesform a pentagon, Whereas elsewhere throughout the pattern the gridtriangles group themselves into hexagons, this being one of thedistinguishing characteristics of .three-way grid construction.

My present invention arises in the discovery that when perfectly flatrectangular sheets are shingled together in a three-way grid pattern andare fastened together Where they overlap in the areas of the geodesiclines of the pattern, a new phenomenon occurs: there are induced t ineach flat rectangular sheet, elements of live cylindrical strutsdefining two triangles of the grid edge to edge in diamond pattern. Theeffect is to produce a three-Way 2 what we may for simplicity term aself-strutted geodesic ply'clome. The hat sheets become inherentgeodesic; they become both roof and beam, both wall and column, and ineach case the braces as Well. They become the Weatherbreak and itssupporting frame or truss all in one. The inherent three Way grid ofcylindrical struts causes ythe structure as la whole to act almost as amembrane in absorbing and distributing loads, and results in a moreuniform stressing of all of the sheets. The entire structure is skinstressed, taut and alive. Dead weight is virtually non-existent.Technically, we say that the structure possesses high tensile integrityin a discontinuous compression system.

Description With reference to the accompanying drawings, I shall nowdescribe the "best mode contemplated by me for carrying out myinvention.

Fig. l is a perspective view of `a geodesic plydome embodying myinventionin a preferred Iform.

` Fig. 2 is a detail perspective view of a portion o-f the Fig. 1construction overlaid upon a diagrammatic representation of a three-Waygrid `as an imaginary projec- -tion of the induced strutting of thedome. The area comprised is representative of one full face of .theicosa with adjacent one-third sectors vof adjacent faces. Combining theone-third-sectors lying at each side of the respective meeting edges ofthe 'adjacent faces, We get three arge diamonds; and

Fig. 3 is an enlarged detail view of the sheets which go to make up oneof these large diamonds. Here the sheets are shown as they would appearwhen laid out flat and before they rare fastened together.

` Fig. 4 is a view similar yto Fig. 3. Imagine that this big diamond isnow a part of the completely assembled dome, and notice how thestructure has inductively produced ve struts in each of the sheets.

Figs. 5 to S, inclusive, show icosa segments of several modiedconstructions in which pyramidal groupings of the triangular grid facesdefined by the induced struts produce in and out convolutions of thespherical surface. In these several constructions the apexes of thepyramids define one sphere and the bases of the pyramids another. Whichof the two spheres is the larger depends on Whether the apexes of thepyramids project outwardly or extend inwardly. The sides of the pyramidsmay ybe regarded as struts connecting elements of the -inner and outerspheres and thus creating a truss.

Fig. 5 follows the `same sheet arrangement as in Figs. 1-4. Because ofthe convoluted, or involute-evolute construction, we get hexagonal andpentagonal pyramids (pentagons at the vertexes of the icosa), which forsimplicity I term a Bhexpent coniiguration. Here the apexes of the hexes'and pents project outwardly (or upwardly from the plane of thedrawing). Notice that a strut is induced along the short axis of eachsheet.

Fig. 6 shows a modified hexpent pattern in which the sheets toe in tothe apexes of the pyramids.

Fig. 7, like Fig. 5, has the same sheet arrangement as in Figs. l-4.Here the induced geodesic triangles of the sheets form an invertedtetrahedron at the center of each face of the icosa, and one of theinduced struts extends the long way of each sheet.

Fig. 8 has a sheet arrangement which may be compared to that of Fig. 6,but with one of the struts extending the long way of the sheet there isformed a pattern of inverted hexpent pyramids.

The construction shown in each of Figs. 5 to S inclusive may be turnedinwardly or outwardly. For example if we think of Fig. 5 as representingthe outer surface of a dome, we have pyramids projecting outwardly withtheir apexes in an outer sphere and their bases (or the corners of theirbases) in an inner sphere. Or if We Yicosa triangle.

'large diamond 'toward its edges.

different` sheet markings, or types of sheet.

think of Fig. 5 as representing the inner surface of a dome, we havepyramids extending inwardly with their apexes in an inner sphere andtheir bases (or the corners of their bases) inQan' outer sphere.

ledges of theadjacent' faces. Thuse area OSVT combinesafoheethird-fsector-of -icosa triangleRST, namely the sector OST withaone-third sector TSV of the adjacent l call ithe combined `sector areaslarge diamonds. i It is' helpfulto see the'largefdiainondswhen`analyzing thestructure'as a whole, because, -once the eye becomespracticed at picking them out,both `thepattern of the icosa faces and ofthe induced three-way strutting is 'more easily discerned. This isespecially so -in lthe 'cases of Figs. 1 4, r Fig. 5 and Fig. 7, in eachofwhich all of the sheets are arranged approximately parallel to themajor axis of the large diamond. This Vbrings the major axes into focus,outlining the icosa faces. Then the eye finds the center of the icosaface, further identiliableV by the small triangular opening at O,surrounded by a series of kite-shaped openings at the meeting edges ofadjacent large diamonds and by square' openings at the edges of theicosa triangle. It is suggested that a brief study of these'characteristic formations with reference to Fig. vl

will be of much help in acquiring a'general grasp ofthe v geodesicalignment of the sheets themselves,and later vof the induced geodesicthree-way grid struttingacross -the corners and centers of each sheet. u

Now, if we are." proceeding by the graphic solution method, we firstlayout the-icosa faces on a scale model sphere, then divide the edges ofone of the `faces into the desired number of equal parts, ormodules,whichdetermines what I call the frequency of the 'three-way grid. Forexample, inFig. 2 I have shown the dot and dash line ST divided into sixymodules numbered l to 6 for identiiication, providing a six-frequencygrid. With the three edges of -thelic'osa -face so' divided it isnecessary only to 1 join each point of one edge'with every ysecondpointon pattern of rsubstantially equilateraly triangles. Now we lay outthefsheets' on the grid pattern as shown in Fig. 2, centering `the shortaxisof each sheetV on alternate grid lines-and working -outwardly fromthe major axis of a With the frequency of six we get irst a row of threesheets in spaced end-to-end arrangement, a row of two sheets ateither'side of this,

and overlapping at the corners, and finally a vsingle sheet coming up tothe center Vof veach icosa face, as clearly shown in` Fig. 2.

Notice that the longitudinal centerlines of the sheets (see therepresentative centerlines a and bin Fig. 2) -lie substantiallyalongVgreat circles of the-sphere, or lie substantially ingreat circle planeswhose intersections with'a common sphere `formgrids `comprisingsubstantially equilateral triangles. The sheets-are nowmarked forinterconnection along the klines of the three-way grid previously laidout. These lines of interconnection will be found` to be substantiallynormal to the aforesaid intersections. Thus the lineof'interconnectionmarked onthe three-way grid at a is normal tocenterline intersection aand that-marked at bv normal to b, etc. It willbe found that the'markings for'interconnection of the sheets willvaryf1'rom'oneA sheet to another depending uponits position in thepattern. Thenumber of diiferentsheet markings depends'upon the`frequency of the grid. With the frequency of six shown in Figs. l-4therevvill be three It is desirable to label,` or color-codethesheets toshow how they are to be put together.

351portions yof the shaded'areas. `In"Fig.4l the effecthas beenconsiderably exaggeratedin order to bring out'ithe Vpoint. Thefself-'struttingphenomenontakes place during assemblyof 'the sheetsaccording` to their coding'ta'nd fastenings, following the designs laidout as above. Such perforations are shown bythe black dots in a sheet atthe lower right of Fig. 3. Notice that additional perforations areprovided near the corners of the sheets so that the sheets will befastened together both in the areas of the grid lines and also at pointssubstantially removed from said lines. This not only buttons down thecorners of the sheets, but assists importantly in creating the induced:struts in -the 'completed structure.

Turning nowr to-Fig.V 3, we see the sheets for a large diamond as theywould appear when laid out -flat and before they* are fastened together.`The 'shadedareas at the outer overlapping corners show the amount ofincreased overlap which occurs when'the sheets are brought into positionfor fastening them together. Once they have been brought into positionand fastened, the sinuses 7, 8, etc., between the grid line markingsclose up and the structure-'assumes its'desired spherical form.-Concurrently, there are induced'in eachlflatv rectangular sheet,elements of ve cylindrical struts dening two triangles of the geodesicgrid edge to edge in diamond pattern.

-Asshown'in Fig. 4, four of these struts cross the corners 'ofthe'sheet' andthe ifthy extends the-short way `of the 'sheetto form the baseof the two Vgeodesic triangles.

These struts, as may be 1discerned from 'theshadng, iare tentv ofoverlap of the sheets, thickness vof 'the `sheetsand "possibly otherfactors. `In some cases the radius of the -bendmay be so4 large that thestruttingis not clearly "visible;or is"perhaps only /visiblejto'apracticedV eye. I

'hav'ehad the draftsrnan try to' simulate -vthephotographic appearanceofthe particular dome represented in Fig. l,

where the geodesic strutting shows u p in the high-lighted fastening'them together in the designated areas of the grid lines andat theircorners kas markedforfactory- "drilled for the' fastenings. When Fig. 4lis'imagined asa 'part of the completely assembleddomd'a comparisonofFigs. 3 and 4 will helpto give an'idea ofthe inductive struttingaction.Fig.'3 is a lstatic assembly of related parts'which know the three-way-geodesic grid' pattern of 'the ficosa; Fig. 4 a dynamicresolution lofthe'pattern "into (a) 'spherical form, (b) with inherent struts express-'ing Vthe pattern in' terms of gentle bendsk in the'sheets,

each bendcomprising elements of a cylindrical surface. It 'seemsremarkable that the bends locate themselves, at

least in part, even in the double thickness' of the overlapping cornersof the sheets Where it might have been supposed that the stiffness ofthe double thickportions would suggest a greater resistance to bending.This resultimplies strongly that the inherent structuring ofthe geodesicgrid pattern is so'natural and strong'in'its tendency to produce aperfect self-supporting sphere that it departs from behaviorpatternspredicted fromordinary -principles of mechanics and strength ofmaterials. 5 Since Ithe behavior of the system as a whole is unpredictedlfrom 'its parts,we say that the resulting structure is synergetic.Such"structures are vastly stronger, `pound"=for "pound, "than anyheretofore known.

The curve of the bends in the sheets, variable according'to the vfactorsnamed in 'the preceding paragraph,l

may comprise elements of a circular cylind'eror elements -of acylinderof varying radius. This is-to say, the radius *ofcurvatureofaparticular strut neednot be uniform. vTosorne degree this 4factor mayvbeifinfluenced by the leverage imposed by-the overlapping areas wherethe vvv`side ofthe geodesic-line, but the strut will in every caseremain substantially a true geodesic linein the sensethat its `axis willliel in aiplanewhse intersection with `a sphere is an element` of agreatcirc1e. The strut itself becomes a chord of that sphere. v

To keep the drawings clear and readable, the fastenings have beenomitted, except as the holes for them have beendepicted in Fig. 3 and asthe geodesic grid lines used in locating them are shown in Figs. l2 'and3. The fastenings themselves may be of any conventional type, and insome constructions it would befeasible to use adhesive means for holdingthe sheets together in the same geodesic alignment.

The sheets may be` of anydesired material, such as plywood, aluminum,steel, plastics, plastic-coated wallboard, composites of plywood andaluminum, plywood and aluminum sheet or foil, etc. I have found thatmarine plywood `in standard ,sheet sizes has ,excellentcharacteristicsfor induced strutting.

If desired, the openings between the sheets can be closed up, this beingmerely a function of the selected frequency of the grid in relation tosheet size. The proportions of the sheets also are subject to variation,but I recommend adherence to substantially a three to live ratio betweenwidth and length as giving best results for most building purposes. Itis even possible to use sheets of other forms than rectangular, but anessential advantage of my construction is that it permits the use ofplain rectangular sheets which are so readily available, stack socompactly for shipment and are least expensive. If the openings betweenthe sheets are not closed up by the boards themselves, they may be usedas skylights, and I have had good results with the use of thin skins oftransparent mylar plastic for covering the openings. In some cases itmay be desired to use an overall plastic inner or outer lining toweatherproof the dome; or weatherproofing may be secured by sealing thejoints with plastic compounds or tape, and painting. Also, theoverlapping of the sheets one upon another can be arranged so that theentire structure is weathershingled to shed water outwardly anddownwardly over the surface of the dome. Such shingling of the sheetscan also be arranged to cover the openings where they come together, oradditional sheets can be slipped in to shingle over the openings.

By laying out the three-way grid pattern so that the radial lines of thepolygons are longer than the lines forming the sides of the bases of thepolygons, we obtain the hexpent and tetrahedral forms of themultiple-sphere trussed constructions explained Ain the outlinedescription of Figs. 5 to 8, inclusive q.v. Thus in Fig. 5 we have, in afour-frequency grid design, a typical hexagonal pyramid with its apex atthe center of the icosa face RST, this pyramid being formed by threesheets, and the two induced triangles of each sheet making two sides ofthe pyramid. Pentagonal pyramids occur at each vertex of the icosa. Thepattern is one comprising hexagonal and pentagonal pyramids the apexesof which define an outer sphere, and the corners of the bases of whichdefine an inner sphere.

In Fig. 6, we again have a pattern of hexagonal and pentagonal pyramids,but here six sheets toe in to the apex of a hexagonal pyramid and fivesheets toe in to the apex of a pentagonal pyramid. In both this view andFig. 5, one of the induced struts extends the short way of the sheet andin this respect there is a similarity to the neutral, or one-sphere,form of Figs. l-4.

In Fig. 7, the induced geodesic triangles of the sheets form an invertedtetrahedron at the center of the icosa face, and one of the inducedstruts of each sheet extends the long way of the sheet to form thecommon base of the two induced triangles. Each triangle is two frequencymodules wide, one frequency module high.

In Fig. 8, we again have a pattern of hexagonal and pentagonal pyramids,six sheets toeing in to the apex of a hexagonal pyramid and ve sheetstoeing in to the apex .of a kpentagonal pyramid, and one of the inducedstruts extending the long'way of the sheet. 1 v

The terms and expressions which I have employed are used in adescriptive and not a limitingsense, and I have no intention ofexcluding such equivalents of the invention described, or of portionsthereof, as fall with- ,in the scope of the claims.

`spherical triangles, said sheets being initiallyV flat and marked forinterconnection' along lines substantially normal to said intersections,said framework 4being characterized by the fact that the sheets arefastened together in the areas of said lines and cylindrical struts areinduced in the sheets defining two geodesic triangles in each sheet.

2. A building framework characterized as defined in claim l, in whichsaid interconnected sheets are rectangular in form.

3. A building framework characterized as defined in claim 1 in whichsaid interconnected sheets are fastened together also at pointssubstantially removed from said lines.

4. A building structure of generally spherical form comprisingoverlapping sheets arranged in a geodesic three-Way grid pattern, saidsheets being initially flat and arranged for interconnection along linesnormal to the lines of the grid pattern, the sheets being fastenedtogether in the areas of the lines of interconnection and cylindricalstruts being induced in the sheets defining two geodesic triangles ineach sheet.

5. A building structure of generally spherical form comprisingrectangular sheets arranged in a geodesic three-way grid pattern on thefaces of a spherical icosahedron with the sheets overlapping at theircorners and fastened together at the overlaps and having inducedcylindrical struts defining two geodesic triangles forming a diamond ineach sheet.

6. A building structure according to claim 5, in which the arrangementof the sheets on the faces of the spherical icosahedron is this: in thediamond formed by geodesic lines joining common vertexes of a commonside of adjacent spherical faces with the centers of said adjacentfaces, the sheets are arranged in parallel rows aligned with the majoraxis of said diamond.

7. A building structure according to claim 6, in which the inducedgeodesic triangles of the sheets form a pattern of hexagonal adnpentagonal pyramids the apexes of which define an outer sphere and thecorners of the bases of which define an inner sphere.

8. A building structure according to claim 6, in which the inducedgeodesic triangles of the sheets form a pattern of inverted hexagonaland pentagonal pyramids the corners of the bases of which define anouter sphere and the apexes of which define an inner sphere.

9. A building structure according to claim 6, in which the inducedgeodesic triangles of the sheets form an inverted tetrahedron at thecenter of each face of the spherical icosahedron and one of the inducedstruts of each rectangular sheet extends the long way of the sheet toform the common base of said two geodesic triangles.

l0. A building structure according to claim 6, in which the inducedgeodesic triangles of the sheets form a tetrahedron at the center ofeach face of the spherical icosahedron and one of the induced struts ofeach rectangular sheet extends the long way of the sheet to form thecommon base of said two geodesic triangles.

l1. A building structure according to claim 5 in which the geodesictriangles of the sheets form a pattern of hexagonal and pentagonalpyramids, six sheets toeing in *Fuller -AJune 29,11954 UNITED STATESPATENT OFFICE CERTIFICATE OE CORRECTION Patent No.- 2,905,113 September22, 1959 Richard Buckminster Fuller It is hereby certified that errorappears in the printed specification of' the above numbered patent.requiring correction and that the said Letter s -f Patent should readIas corrected below.

Signed and sealed this 22nd day of March 1960.,

(SEAL) Attest:

KAEL H. AXLTNE ROBERT c. WATSON Attesting OHicer Commissioner of Patents

